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5/1/2020
1
Advanced Computation:
Computational Electromagnetics
Transfer Matrix Method Using Scattering Matrices
Outline
• Calculating reflected and transmitted power
• Simplifications for 1D transfer matrix method
• Notes on implementation
• Block diagram
Slide 2
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Slide 3
Calculating Transmitted and Reflected Power
Recall How to Calculate the Source Parameters
Slide 4
inc 0 inc
sin cos
sin sin
cos
k k n
0
ˆ ˆ 0
1zn a
incTE
inc
ˆ 0
ˆˆ 0
ˆ
ya
n ka
n k
inc TETM
inc TE
ˆˆ
ˆ
k aa
k a
Incident Wave Vector Surface Normal Unit Vectors in Directions of TE & TM
Composite Polarization Vector
TE TMTE TMˆ ˆP p p aa
Right‐handedcoordinate system
1P
In CEM, the magnitude is usually set to 1.
ˆxa
ˆya ˆza
Unit vectors along x, y, and z axes.
Can be any direction in the x‐y plane
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Solution Using Scattering Matrices
Slide 5
The external fields (i.e. incident wave, reflected wave, transmitted wave) are related through the global transfer matrix.
globalref inc
trn
c cS
c 0
This matrix equation can be solved to calculate the mode coefficients of the reflected and transmitted fields.
global globalref 11 12 inc
global globaltrn 21 22
c S S c
c 0S S
globalref 11 inc
globaltrn 21 inc
c S c
c S c
1inc ref
x
y
P
P
c W
rightinc not typically usedc
x
y
z
P
P P
P
𝑃 and 𝑃 are obtained from the polarization vector �⃗�.
Note that 𝑃 is not needed.
Calculation of Transmitted and Reflected Fields
Slide 6
The procedure described thus far calculated the mode coefficients cref and ctrn.
The transmitted and reflected fields must be calculated from the mode coefficients.
ref
ref refref
trn
trn trntrn
x
y
x
y
E
E
E
E
W c
W c
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Calculation of the Longitudinal Components
Slide 7
The longitudinal field component 𝐸 must still be calculated on both the reflection and transmission sides.
These are calculated from 𝐸 and 𝐸 using Maxwell’s divergence equation.
0, 0, 0,
0, 0, 0,
0, 0, 0,
0, 0, 0,
0, 0,0,
0
0
0
0
jk r jk r jk rx y z
jk r jk r jk rx x y y z z
x x y y z z
z z x x y y
x x y yz
z
E
E e E e E ex y z
jk E e jk E e jk E e
k E k E k E
k E k E k E
k E k EE
k
ref refref
ref
trn trntrn
trn
x x y yz
z
x x y yz
z
k E k EE
k
k E k EE
k
Note:
0 reduces to
0 when is homogeneous.
E
E
Calculation of Power Flow
Slide 8
2trn
trn r,trn
2 incr,incinc
Re
Re
z
z
E kT
kE
1 materials have loss
1 materials have no loss and no gain
1 materials have gain
R T
Reflectance 𝑅 is defined as the fraction of power reflected from a device.2
ref
2
inc
ER
E
2 22 2
x y zE E E E
Transmittance 𝑇 is defined as the fraction of power transmitted through a device.
It is always good practice to check for conservation of power.
Note: These formulas will be derived in a later lecture.
Note: Recall 1A R T
incE P
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Reflectance and Transmittance on a Decibel Scale
Slide 9
Decibel Scale
dB 1020 logP A
dB 1010 logP P
How to calculate decibels from an amplitude quantity A.
How to calculate decibels from a power quantity P.
2 2dB 10 10 10 log 20logP A P A A
Reflectance and Transmittance
Reflectance and transmittance are power quantities, so
dB 10
dB 10
10 log
10log
R R
T T
Slide 10
Simplifications for1D Transfer Matrix Method
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Analytical Expressions for 𝐖 and 𝛌
Slide 11
Using this relation, the matrix equation for 2 can be greatly simplified.
2zk I
Since 2 is a diagonal matrix, it can be concluded that
2 2
1 0
0 1
W I
λ Ω
The dispersion relation with a normalized wave vector is2 2 2
r r x y zk k k
0
0
zz
z
jkjk
jk
λ I
0
0
z
z
jk zz
jk z
ee
e
λ
1 0 identity matrix
0 1
I
For isotropic materials and diagonally anisotropic materials, it is not necessary to actually solvethe eigen‐value problem to obtain the eigen‐modes!
A lot of algebra
2 2r r r r2
2 2r r r r r r
1 x y x x y x
y x y y x y
k k k k k k
k k k k k k
Ω PQ
2
2
0
0
z
z
k
k
Simplifications for TMM in LHI Media
Slide 12
Given that
1 0
0 1i
W I ,i z ijkΩ I i iλ Ω
The expression for the eigen‐vectors for the magnetic fields V reduces to
1 1i i i i i i
V Q Wλ QΩ
When calculating scattering matrices, the intermediate matrices Ai and Bi reduce to
1 1 1g g g
1 1 1g g g
i i i i
i i i i
A W W V V I V V
B W W V V I V V
The fields and mode coefficients are now the same thing!
ref trn1
inc ref ref 11 inc 11 inc trn 21 inc 21 incref trn x x x x
y y y y
P P E E
P P E E
c W W S c S c W S c S c
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Simplified External S‐Matrices in LHI Media
Slide 13
The reflection‐side scattering matrix reduces to
ref 111 ref ref
ref 112 ref
ref 121 ref ref ref ref
ref 122 ref ref
2
0.5
S A B
S A
S A B A B
S B A
1ref g ref
1ref g ref
A I V V
B I V V
trn 111 trn trn
trn 112 trn trn trn trn
trn 121 trn
trn 122 trn trn
0.5
2
S B A
S A B A B
S A
S A B
1trn g trn
1trn g trn
A I V V
B I V V
The transmission‐side scattering matrix reduces to
,I
,I
r
r
r,g
r,g
0limL
,II
,II
r
r
r,g
r,g
refs
trns
0limL
Slide 14
Notes on Implementation
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Outline
Slide 15
• Step 0 – Define problem
• Step 1 – Dashboard
• Step 2 – Describe device layers
• Step 3 – Compute wave vector components
• Step 4 – Compute gap medium parameters
• Step 5 – Initialize global scattering matrix
• Step 6 – Main loop through layers
• Step 7 – Compute reflection side scattering matrix
• Step 8 – Compute transmission side scattering matrix
• Step 9 – Update global scattering matrix
• Step 10 – Compute source
• Step 11 – Compute reflected and transmitted fields
• Step 12 – Compute reflectance and transmittance
• Step 13 – Verify conservation of power
• Compute P and Q
• Compute eigen‐modes
• Compute layer scattering matrix
• Update global scattering matrix
Step 6: Iterate through layers
human does this
computer does the rest
Storing the Problem
Slide 16
How is the problem stored and described in the dashboard of TMM?
Note that this is a right‐handed coordinate system.
ˆ ˆ ˆx y za a a
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Storing the Problem
Slide 17
How is the problem stored and described in the dashboard of TMM?
Device is described in three 1D arrays.
ER = [ 2.50 , 3.50 , 2.00 ];UR = [ 1.00 , 1.00 , 1.00 ];L = [ 0.25 , 0.75 , 0.89 ];
Storing the Problem
Slide 18
External materials:
er1, er2, ur1 and ur2
How is the problem stored and described in the dashboard of TMM?
Device is described in three 1D arrays.
ER = [ 2.50 , 3.50 , 2.00 ];UR = [ 1.00 , 1.00 , 1.00 ];L = [ 0.25 , 0.75 , 0.89 ];
r1
r1
r2
r2
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Storing the Problem
Slide 19
External materials:
er1, er2, ur1 and ur2
How is the problem stored and described in the dashboard of TMM?
Device is described in three 1D arrays.
ER = [ 2.50 , 3.50 , 2.00 ];UR = [ 1.00 , 1.00 , 1.00 ];L = [ 0.25 , 0.75 , 0.89 ];
The source:
theta, phi, pte, ptm, and lam0
r1
r1
r2
r2
Storing the Problem
Slide 20
How is the problem stored and described in the dashboard of TMM?
r1
r1
r2
r2
After simulation!
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Storing Scattering Matrices
Slide 21
The scattering matrix S is often talked about as a single matrix.
11 12
21 22
S SS
S S
However, the scattering matrix S is rarely used this way. Most commonly, the individual terms S11, S12, S21, and S22 are handled separately.
11 12
21 22
S SS
S S 11 12 21 22 , , , and S S S S
So, scattering matrices are best stored as the four separate components of the scattering matrix.
Initializing the Global Scattering Matrix
Slide 22
What are the ideal properties of nothing?
2. Does not reflect at all.
global global12 21 S S I
global global11 22 S S 0
Therefore, the global scattering matrix is initialized according to
global
0 IS
I 0This is NOT an identity matrix!Look at the position of the 0’s and I’s.
Before iterating through all the layers, the global scattering matrix must be initialized as the scattering matrix of “nothing.”
1. Transmits 100% of power with no phase change.
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Calculating the Parameters of the Gap Media
Slide 23
2 2 2,g r,g r,g
g
z x yk k k
W I
The analytical solution for a homogeneous gap medium is
2r,g r,g
g 2r,g r,g r,g
1 x y x
y x y
k k k
k k k
Q
Any choice of r,g and r,g is possible. However, It is best to avoid the case of kz,g = 0.
2 2r,g r,g1.0 and 1 x yk k
Given this choice, the parameters reduce to
2
g 2
1
1
x y y
x x y
k k k
k k k
Q
g
g gj
W I
V Q
W is not even used in isotropic TMM!
g ,g
1g g g
zjk
λ I
V Q λ
To do this in a mathematically convenient way, choose
Mode Coefficients = Fields (Sometimes)
Slide 24
Recall that the fields are calculated from the mode coefficients through
This means that in TMM, the mode coefficients are the fields.
inc ref trn
inc inc ref ref trn trninc ref trn x x x
y y y
E E E
E E E
W c W c W c
However, for TMM using LHI materials, 𝐖 𝐈 always.
inc ref trn
inc ref trninc ref trn x x x
y y y
E E E
E E E
c c c
This is a special thing for TMM with LHI materials. It does not hold for other methods or when anisotropic materials are being simulated.
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WRONG
Calculating 𝐗 exp 𝛀 𝑘 𝐿
Slide 25
0
0
0
0
0
z i
i i
z i
jk k Lk L
i jk k L
ee
e
ΩX
Recall the correct answer:
It is incorrect to use the function exp() directly on the matrix 𝛀 because in MATLAB exp()calculates a point‐by‐point exponential, not a matrix exponential.
X = exp(OMEGA*k0*L);X =
0.0135 + 0.9999i 1.00001.0000 0.0135 + 0.9999i
Approach #1: expm() Approach #2: diag()
X = expm(OMEGA*k0*L);
X =0.0135 + 0.9999i 0
0 0.0135 + 0.9999i
X = diag(exp(diag(OMEGA)*k0*L));
X =0.0135 + 0.9999i 0
0 0.0135 + 0.9999i
Efficient Calculation of Layer S‐Matrices
Slide 26
There are redundant calculations in the equations for the scattering matrix elements.
111 22
112 21
11
11
i ii i i i i i
i ii i i i
i i i i i i
i i i i i ii
A X B A X B
A
S
X B A
S X B A X A B
S S X A B AB BX
These are more efficiently calculated as
0
1g
1g
1
1 111 22
1 112 21
i i
i i
i i
k Li
i i i i i i
i ii i i i i i
i ii i i i i
e
λ
A I V V
B I V V
X
D A X B A X B
S S D X B A X A B
S S D X A B A B
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Efficient Star Product
Slide 27
After observing the equations to implement the Redheffer star product, some common terms can be identified. Calculating these multiple times is inefficient so they should be calculated only once.
1B A B
21 22 11
1B A B
2
AB A B A11 11 11 21
AB B12 12
AB A21 21
AB B A B
1A B A
12 11 22
1
2
A B A12 1
21 2 1122 22 2 1
1 22
2
S
S
S I S S
S I S
S
S S S
I
S S
S
S S S
S S
I S S
S
S
S
1A B A
12 11 2
1B A B
21 22 1
2
1
F
D
S I S S
S I S S AB A B S S S
A B A11 11 11 21
B12 12
A21 21
B A B22 22 22 12
AB
AB
AB
AB
S S S S
S S
S S
S S SF S
F
D
D
Using the Redheffer Star Product as an Update
Slide 28
The global scattering matrix is updated using a Redheffer star product.
Close attention MUST be paid to the order that the equations are implemented so that values are not overwritten.
1global
12 11 22
1global global
21 22 11
global global22 22 22
global21 21
global global12 12
global global11 11
global1
1
2
1 21
i i
i
i
i
i i
D S I S S
F S I S S
S S FS
S FS
S DS
S S DS S
S
1global global
12 11 22
1global
21 22 11
global global global11 11 11 21
global12 12
global global21 21
global global22 22 22 12
i
i i
i
i
i i
D S I S S
F S I S S
S S DS S
S DS
S FS
S S FS S
global globali S S S global global i S S S
reverse order
standard order
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Can TMM Fail?
Slide 29
Yes!
The TMM can fail to give an answer and behave numerically strange any time kz = 0. This happens at a critical angle when the transmitted wave is at or very near its cutoff.
This problem was prevented in the gap medium, but this can also happen in any of the physical layers or in the transmission region under the right conditions.
2 2r r x yk k
This happens in any layer where
Slide 30
Block Diagram of TMM
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Calculate Transmittance and Reflectance
2
ref
trn2 r,trn
trn incr,inc
Re
Re
z
z
R E
kT E
k
Finish
Calculate Longitudinal Field Components
ref refref
ref
trn trntrn
trn
x x y yz
z
x x y yz
z
k E k EE
k
k E k EE
k
Calculate Transmitted and Reflected Fields
ref
ref 11 srcref
trn
trn 21 srctrn
x
y
x
y
E
E
E
E
e S e
e S e
Calculate Source
TE TE TM TM
src
ˆ ˆ
1
x
y
P p a p a
P
P
P
e
yes
Connect to External Regions
global ref global
global global trn
S S S
S S S
Done?
no
Loop through all layers
Update Global Scattering Matrix
global global
1global global12 11 22
1global21 22 11
global global global11 11 11 21
global12 12
global global21 21
global global22 22 22 12
i
i
i i
i
i
i i
S S S
D S I S S
F S I S S
S S DS S
S DS
S FS
S S FS S
Block Diagram of TMM Using S‐Matrices
Slide 31
Calculate Scattering Matrixfor Layer i
01g
1g
1
1 111 22
1 112 21
i ik Li i i
i i
i i i i i i
i ii i i i i i
i ii i i i i
e
λA I V V X
B I V V
D A X B A X B
S S D X B A X A B
S S D X A B A B
Initialize Global Scattering Matrix
global
0 IS
I 0
Start
Calculate Parameters for Layer i
2 2,
2
2
1,
1
z i i i x y
x y i i x
ii y i i x y
i z i i i i
k k k
k k k
k k k
jk
Q
Ω I V QΩ
Calculate Gap Medium Parameters
2
g g g2
1
1
x y y
x x y
k k kj
k k k
Q V Q
Calculate Transverse Wave Vectors
inc
inc
sin cos
sin sin
x
y
k n
k n
How to Handle Zero Number of Layers
Slide 32
Follow the block diagram!!
Setup your loop this way…
NLAY = length(L);for nlay = 1 : NLAY
...end
If NLAY = 0, then the loop will not execute and the global scattering matrix will remain as it was initialized.
global
0 IS
I 0
For zero layers:
ER = [];UR = [];L = [];
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